Notice that \(N(x) = \lbrace -\mathbf {\nu }_{K}^{\phi }(x)\rbrace  \)  by [1]} and the strict convexity of
\(\phi \) . We use [2]} to infer that
\(\phi ^{\prime }(\,\cdot \,; \alpha ) \)  is an upper semi-continuous function and to
estimate
\(\sup \lbrace -\phi ^{\prime }(v, -\operatorname{grad}\delta _{K}^{\phi }(x)) : v \in N(x,\varepsilon ) \rbrace  \le \sup \lbrace \phi ^{\prime }(v, \operatorname{grad}\delta _{K}^{\phi }(x)) : v \in N(x,\varepsilon ) \rbrace  < 0\) 
